In our lecture we have proven the following:
Lemma. Let $\Phi: V \to \mathbb{R}$, where $V$ is a real (reflexive) Banach space, possess the Gâteaux derivative $A: V \to V^*$. Then $\Phi$ is convex iff $A$ is monotone.
We call an operator pseudomonotone if $u_n \rightharpoonup u$ in $V$ and $\limsup_{n \to \infty} \langle A u_n, u_n - u \rangle$ implies that $\langle Au, u -w \rangle \le \liminf_{n \to \infty} \langle A u_n, u_n - w \rangle$ holds for all $w \in V$.
Is there a notion of pseudo convexity (I understand there are multiple definitions) such that we can write pseudomonotone instead of monotone and pseudoconvex instead of convex?